Text of Geometric Sequences and Series A sequence is geometric if the ratios of consecutive terms are the...
Geometric Sequences and Series A sequence is geometric if the
ratios of consecutive terms are the same. 2, 8, 32, 128, 512,...
Definition of Geometric Sequence geometric sequence The common
ratio, r, is 4.
The nth term of a geometric sequence has the form a n = a 1 r n
- 1 where r is the common ratio of consecutive terms of the
sequence. The nth Term of a Geometric Sequence 15, 75, 375,
1875,... a 1 = 15 The nth term is 15(5 n-1 ). a 2 = 15(5) a 3 =
15(5 2 ) a 4 = 15(5 3 )
Example 3. Find the 15 th term of the geometric sequence whose
first term is 20 and whose common ration is 1.05.
Example 4. Find a formula for the nth term of the following
geometric sequence. What is the ninth term of the sequence? 5, 15,
45, Find the common ratio
The sum of the first n terms of a sequence is represented by
summation notation. Definition of Summation Notation index of
summation upper limit of summation lower limit of summation
The Sum of a Finite Geometric Sequence The sum of a finite
geometric sequence is given by 5 + 10 + 20 + 40 + 80 + 160 + 320 +
640 = ? n = 8 a 1 = 5
Write out a few terms. If the index began at i = 0, you would
have to adjust your formula
Definition of Geometric Series The sum of the terms of an
infinite geometric sequence is called a geometric series. a 1 + a 1
r + a 1 r 2 + a 1 r 3 +... + a 1 r n-1 +... If |r| < 1, then the
infinite geometric series has the sum
Example 7. Use a graphing calculator to find the first six
partial sums of the series. Then find the sum of the series. Use
the formula for the sum of an infinite series to find the sum.
Example 8. Find the sum of 3 + 0.3 + 0.03 + 0.003 + ,
Example 9. A deposit of $50 is made on the first day of each
month in a savings account that pays 6% compounded monthly. What is
the balance at the end of 2 years? This type of savings plan is
called an increasing annuity. The first deposit will gain interest
for 24 months, and its balance will be The second deposit will gain
interest for 23 months The last deposit will gain interest for only
one month The total balance will be the sum of the balances of the